The musical scales are A B C D E F G A, or Do Re Mi Fa So La Ti Do. If you do it that way, yes, there are eight. But the last note is the same as the first but at a higher pitch. If this were math, we’d basically be dealing with a base-7 system, with A or Do playing the part of zero.
So why is the set of scales called an octave if it’s all based on a base-7 system?
EDIT: Many of the first answers from when I originally asked were helpful but now I’m getting a lot of wrong answers from people who don’t seem to understand how numbers work. In a base-8 system, there are eight unique numbers, 0 through 7, after which it goes to 10. If you translate the notes into numbers, you don’t get 0 through 7, though. You get 0 through 6, after which it goes to 10 at the second Do. That’s why I was trying to reconcile that with the term “octave.”
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The set of all the notes is not called an octave, it is called a diatonic scale. What we understand in Western music to be scales are all some form of a diatonic scale(major, minor, mixolydian etc.). All diatonic scales consist of 7 intervals, 5 whole steps and 2 half steps. If you think of the scale as a fence, there are 8 posts(notes), with 7 rails(intervals) in between. You need to have the octave to know the length of the 7th interval. Since the octave above the root is the eighth note, its name reflects that.
FWIW, there are 8 interval scales, such as the octatonic scale, which consists of 4 whole steps and 4 half steps. In this scale, the octave is actually the 9th note. These are not used traditionally and would sound pretty jarring to most ears.
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